Derivation of multiple but finite number of imaginary eigenvalues for a two-layer diffusion-reaction problem

Abstract

Diffusion-reaction problems occur commonly in heat and mass transfer analysis. In the particular case of a multilayer body under certain conditions, such problems are known to result in imaginary eigenvalues that indicate divergence in the temperature/concentration field at large times. For small values of the reaction coefficient, past work has derived conditions in which an imaginary eigenvalue may exist. This work generalizes this result by showing that more than one, but not infinite imaginary eigenvalues may exist under certain conditions in a one-dimensional two-layer diffusion-reaction problem. A systematic investigation of the eigenequation in the imaginary space is carried out in order to derive a closed form expression for the number of imaginary eigenvalues for any given problem. It is shown that, for other parameters being fixed, the larger the values of the reaction coefficients, the greater is the number of imaginary eigenvalues. However, unlike real eigenvalues, it is shown that the number of imaginary eigenvalues is never infinite. While presented for a two-layer body, similar techniques may be applied for a more general multilayer body, although the derivation of explicit equations may be more challenging. Besides its theoretical importance, accounting for imaginary eigenvalues in the problem discussed here is clearly important to ensure accuracy of the mathematical model. The present work contributes towards this by systematically identifying the number and nature of imaginary eigenvalues that occur in multilayer diffusion-reaction heat and mass transfer problems.